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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A rank theorem for infinite dimensional spaces


Author: J. P. Holmes
Journal: Proc. Amer. Math. Soc. 50 (1975), 358-364
MSC: Primary 58C25
MathSciNet review: 0383452
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Abstract: Suppose $ X$ is a Banach space, $ U$ is an open set of $ X$ containing 0, and $ f$ is a continuously differentiable function from $ U$ into $ X$ satisfying $ f(0) = 0$ and $ f'{(0)^2} = f'(0)$. An additional hypothesis is given for $ f$ which, in case $ X$ is finite dimensional, is equivalent to assuming rank $ f'(x) = \operatorname{rank} f'(0)$ for all $ x$ in some neighborhood of 0. Under this hypothesis one obtains a local factorization of $ f$ into $ {h_1} \circ f'(0) \circ {h_2}$ where each of $ {h_1}$ and $ {h_2}$ is a continuously differentiable homeomorphism. In addition there is a neighborhood of 0 in $ {f^{ - 1}}(\{ 0\} )$ which is the image of a continuously differentiable retraction. An application of these results to the theory of differentiable multiplications is given.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0383452-0
PII: S 0002-9939(1975)0383452-0
Keywords: Rank theorem, differentiable semigroups
Article copyright: © Copyright 1975 American Mathematical Society